Surface plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening-length limit

Abstract

According to the hydrodynamic Drude model, surface plasmon resonances of metallic nanostructures blueshift owing to the non-local response of the metal's electron gas. The screening length characterizing the non-local effect is often small relative to the overall dimensions of the metallic structure, which enables us to derive a coarse-grained non-local description using matched asymptotic expansions; a perturbation theory for the blueshifts of arbitrary-shaped nanometallic structures is then developed. The effect of non-locality is not always a perturbation and we present a detailed analysis of the 'bonding' modes of a dimer of nearly touching nanowires where the leading-order eigenfrequencies and eigenmode distributions are shown to be a renormalization of those predicted assuming a local metal permittivity.

Keywords: non-locality; plasmonics; singular perturbations; surface plasmons.

Figure 1.

Normalized absorption cross-section (3.4) for a metal nanosphere (using typical values a=2 nm, β= 10⁶ m s⁻¹, $\hslash {\omega }_{\mathrm{p}}=9$ eV and $\hslash \gamma =0.13$ eV [25,27]) with δ≈0.037 under plane-wave illumination. Black line, local prediction [1]; blue line, non-local prediction (3.2); dashed black and blue lines, local and non-local predictions of resonant frequency, respectively (cf. (3.7)) and dotted red lines, (3.8) for the high-order bulk-plasmon frequencies. (Online version in colour.)

Figure 2.

Multimode excitation of a metal sphere by a nearby radially oriented electric dipole positioned as shown in the inset ($\hslash {\omega }_{\mathrm{p}}=9$ eV, $\hslash \gamma =0.13$ eV and δ=0.01). Black and blue lines, absolute magnitude of the induced radial field at the position of the dipoleaccording to the local and non-local models, respectively. Vertical black and blue dotted lines, first and first two terms of (3.10). (Online version in colour.)

Figure 3.

Dimensionless schematics of nanometallic configurations considered in §4 and §5.

Figure 4.

Dimensionless near field $|\mathrm{Im}[\hat{\mathit{ı}}\mathbf{\cdot }\mathrm{\nabla }\phi ]|$ external to metallic prolate spheroid (γ/ω_p= 0.014, δ=0.04) subjected to a plane wave polarized along $\hat{\mathit{ı}}$, for several values of the slenderness parameter s (the field is probed on the revolution axis at unit distance from the spheroid tip). Black and blue lines are respectively the local and non-local responses from a quasi-static simulation. The vertical dotted black and blue lines, respectively, depict thedipolar resonant frequency predicted from local theory (cf. (4.39)) and our perturbative non-local theory (cf. (4.40)). (Online version in colour.)

Figure 5.

Near-contact asymptotic structure of bonding modes of a circular cylindrical dimer in the local approximation.

Figure 6.

Gap-field distributions E(X) (cf. (5.5)) corresponding to the first three resonance frequencies. Each eigenfrequency is doubly degenerate with eigenfields even and odd in X.The black lines are the ‘local’ distributions (5.19) and (5.20). The blue lines depict the ‘non-local’ renormalized distributions (cf. (5.34)) at the blueshifted frequencies (5.35), for τ=δ/h=0.75. (Online version in colour.)

Figure 7.

Fundamental surface plasmon bonding-mode of a gold circular cylindrical dimer (a=10 nm, $\hslash {\omega }_{\mathrm{p}}=3.3$ eV, β=0.0036c, giving λ=0.215 nm and δ=0.0215). See text for details on the various data andapproximations shown. (Online version in colour.)

Figure 8.

(a) Normalized absorption cross section (3.4) of a nanometallic sphere under plane-wave illumination in the quasi-static limit—same parameters as in figure 1. Black and blue lines, respectively, depict the local-theory prediction (3.2) and the non-local-theory prediction (3.5). Red line depicts the prediction (6.2) of the uniform local-analogue model (6.1).(b) Field enhancement at the centre of a 0.2 nm wide gap between a pair of gold cylinders of radius a=10 nm (h=0.01), under plane-wave illumination with incident field polarized along the line of centres; same material parameters as in figure 7. Black line, solution of local model; red line, solution of the uniform local-analogue model (6.1); symbols, numerical data extracted from Luo et al.[25] with a plot digitizer. (Online version in colour.)